Free probability induced by electric resistance networks on energy Hilbert spaces

Opuscula Mathematica

Ilwoo Cho¹  Palle E. T. Jorgensen² 

[1] St. Ambrose University, Department of Mathematics, 518 W. Locust St., Davenport, Iowa, 52803, USA;The University of Iowa, Department of Mathematics, 14 McLean Hall, Iowa City, Iowa, 52242, USA;

关键词: directed graphs;    graph groupoids;    electric resistance networks;    ERN-groupoids;    energy Hilbert spaces;    ERN-algebras;    free moments;    free cumulants;   

DOI  :  http://dx.doi.org/10.7494/OpMath.2011.31.4.549

来源: DOAJ

【 摘 要 】

We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space \(H_{\mathcal{E}}\). From \(H_{\mathcal{E}}\), one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on \(H_{\mathcal{E}}\). With the use of our ERN-groupoid, we show that \(H_{\mathcal{E}}\) may be derived as a representation Hilbert space of a universal representation of a groupoid algebra \(\mathfrak{A}_G\), and we display other representations. Among our applications, we identify a free structure of \(\mathfrak{A}_G\) in terms of the energy form.

【 授权许可】

Unknown